std dev of APPL is 0.01848158065635535
BUSI 721, Fall 2022
JGSB, Rice University
Kerry Back
\[ \bar{r}_p = w_1 \bar{r}_1 + w_2 \bar{r}_2\]
\[w_1r_1 + w_2r_2 - (w_1 \bar{r}_1 +w_2\bar{r}_2)\]
\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]
\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2) + 2 w_1 w_2 \text{cov}(r_1, r_2)\]
\[\left(\frac{1}{2}\right)^2 100\text{%}^2 + \left(\frac{1}{2}\right)^2 100\text{%}^2 = 50\text{%}^2\]
std dev of APPL is 0.01848158065635535
\[\small var(r_{p})=\sum_{i=1}^n w_{i}^2var(r_i)+2\sum_{i=1}^n \sum_{j=i+1}^n w_{i}w_{j}cov(r_{i},r_{j})\]
There are \(\frac{n(n-1)}{2}\) covariance terms.
\(n\) assets, all variances are the same = \(\sigma^2\),
all correlations are the same = \(\rho\),
all weights = \(\frac{1}{n}\).
\[\small var(r_{p})=\sum_{i=1}^n (\frac{1}{n})^2\sigma^2+2\sum_{i=1}^n \sum_{j=i+1}^n (\frac{1}{n})^2\rho\sigma^2\]
\[\small var(r_{p})=\frac{1}{n}\sigma^2+\frac{n-1}{n}\rho\sigma^2 \rightarrow \rho\sigma^2\]
\[\small (w_{A}\sigma_{A}+w_{B}\sigma_{B})^2=w_{A}^2\sigma_{A}^2+w_{B}^2\sigma_{B}^2+2w_{A}w_{B}\sigma_{A}\sigma_{B}\]
same as \(var_{p}\) except no correlation in the last term. So,
\[(w_{A}\sigma_{A}+w_{B}\sigma_{B})^2>var(r_{p})\]
Compare holding only asset B to holding some mix of A and B.
Substitute \(w_{B}=(1-w_{A})\). Variance in general is
\[\small w_{A}^2\sigma_{A}^2+(1-w_{A})^2\sigma_{B}^2+2w_{A}(1-w_{A})\rho\sigma_{A}\sigma_{B}\]
Derivative with respect to \(w_{A}\) evaluated at \(w_{A}\)=0:
\[\small 2w_{A}\sigma^2_{A}-2(1-w_{A})\sigma^2_{B}+2(1-2w_{A})\rho\sigma_{A}\sigma_{B}=-2\sigma^2_{B}+2\rho\sigma_{A}\sigma_{B}\]
Risk is lowered by \(w_{A}\) \(\uparrow\) when \(\rho\sigma_{A}<\sigma_{B}\)